Explain why some groups yield large-distance two-block codes

Explain why certain finite groups yield two-block group-algebra codes with large distance, particularly in light of the observed concentration of good codes in a small number of groups.

Background

The paper surveys large computational censuses of two-block group-algebra codes and observes that high-quality codes are strongly concentrated in particular finite groups. The authors develop algebraic explanations for some groups producing nonzero logical dimension, but do not identify a corresponding mechanism explaining why some groups produce codes with unusually large distance.

This question is explicitly presented as an open problem for future theoretical work based on the census data.

References

We offer some explanation for why some groups give logical qubits, and offer the open question of why some groups give large distance to the community.

— Bicycle Race: small two-block codes on the Pareto frontier  (2610.13880 - Lee, 9 Oct 2026) in Section 1, Introduction

We might then conjecture a bound that goes like $d \lesssim c \sqrt{n}$ for some constant $c$.

— Bicycle Race: small two-block codes on the Pareto frontier  (2610.13880 - Lee, 9 Oct 2026) in Section 4, “Where the good codes live”

In the end, it seemed like we had just as many frontier codes that were covers as were primitive, and we were unable to obtain any consistent rule for when one outperformed the other.

— Bicycle Race: small two-block codes on the Pareto frontier  (2610.13880 - Lee, 9 Oct 2026) in Section 4, “Where the good codes live”

We could not explain the parity pattern either, which seems to concern $d$ rather than $k$; although, as mentioned the parity pattern may simply be a coarse-grained view of the most concentrated group.

— Bicycle Race: small two-block codes on the Pareto frontier  (2610.13880 - Lee, 9 Oct 2026) in Section 4, “Where the good codes live”